Abstract

Consider an operator T : E ! X(ì) from a Banach space E to
a Banach function space X(ì) over a finite measure ì such that its dual map
is p-th power factorable. We compute the optimal range of T that is defined
to be the smallest Banach function space such that the range of T lies in it
and the restricted operator has p-th power factorable adjoint. For the case
p = 1, the requirement on T is just continuity, so our results give in this case
the optimal range for a continuous operator. We give examples from classical
and harmonic analysis, as convolution operators, Hardy type operators and the
Volterra operator.